A competitive equilibrium of this model economy consists of sequences of allocations {ct, Lt, Kt+1, st, nt,net, It, ηt, νt, Ht}∞t=0, of prices{wt, rkt+1, rt+1, qt}∞t=0and of exogenous processes{zt, ωt}∞t=0 such that (i) the allocations solve the household’s, the firm’s and the financial intermediary’s problems at the equilibrium prices and (ii) markets for factor inputs clear. The following equi-librium conditions must be satisfied:
Ul(t)
Uc(t) =wt (65)
Uc(t) =β(1 +rt+1)EtUc(t+ 1) (66)
rkt+1 = zt+1FK(Kt+1, Ht+1) +qt+1(1−δ)
qt −1 (67)
wt=ztFH(Kt, Ht) (68)
nt=ωtent (69)
qtst= ηt
λ−νt
nt (70)
νt=Et[(1−θ)βΛt,t+1(rkt+1−rt+1) +βΛt,t+1θqt+1st+1 qtst
νt+1] (71)
ηt=Et[(1−θ)βΛt,t+1(1 +rt+1) +βΛt,t+1θnt+1
nt ηt+1] (72)
e
nt+1 =θ[(rkt+1−rt+1) ηt
λ−νt
+ (1 +rt+1)]nt+ǫnt (73)
qtst=qtKt+1 (74)
Kt+1 = (1−δ)Kt+ Φ It
Kt
Kt (75)
qt=
Φ′ It
Kt
−1
(76)
Lt=Ht (77)
Ct+It=ztF(Kt, Ht) (78)
log(zt+1) =ρzlog(zt) +ǫzt+1 (79)
log(ωt+1) =ρωlog(ωt) +ǫωt+1 (80)
Appendix D: Business Cycle Statistics of Aggregate Financial Variables of the whole U.S. Financial Sector
For interested readers, this section documents empirical cyclical properties of aggregate mea-sures of the leverage ratio, debt and equity of U.S. financial firms and of the credit spread using quarterly data for the period 1952-2009. In particular, I compute standard business cycle statis-tics of the aggregate financial variables, such as their standard deviations, cross-correlations with output.
I use quarterly balance sheet data from the Flow of Funds Accounts of the Federal Reserve Board.27 The theoretical model described below treats the entire financial intermediary sector as a group of identical institutions although there is a considerable amount of heterogeneity among financial institutions in terms of both their functions and balance sheet structures. For example, some financial intermediaries such as private pension funds, mutual funds, retirement funds, are financed only by equity while some others such as banks, security-brokers and dealers use leverage extensively. In order to be consistent with the model, I only select financial institutions that always carry some leverage.
I focus on both depository and non-depository financial institutions. The depository in-stitutions are U.S. chartered commercial banks, savings inin-stitutions, and credit unions. The non-depository institutions are issuers of asset-backed securities, bank holding companies, se-curity brokers and dealers, finance companies, insurance companies, funding corporations, and real estate investment trusts. These institutions perform the majority of activity in the U.S.
financial sector as measured by their total assets.28 Liabilities are defined as the sum of “Total liabilities” of each of the aforementioned depository and non-depository financial institutions in the U.S. financial system, whileNet Worth is defined as the sum of “Total financial assets”
minus the sum of “Total liabilities” of the same institutions. Leverage ratio is the ratio of Liabilities to Net Worth. Credit spread measure I use is the difference between quarterly real return to capital and quarterly real deposit rate. Quarterly real return to capital data are taken from Gomme et.al. (2011). Quarterly deposit rate data is taken from Federal Reserve Economic
27Total financial assets and total liabilities in the Flow of Funds Accounts are partly measured at book values and may be different from market values. The differences between book values and market values are more likely to disappear when the balance sheet of a particular financial institution is marked to market and/or when total financial assets or liabilities are short-term.
28The total assets of these institutions is 90% of the total assets of the U.S. financial sector. Moreover, our definition of U.S. financial sector includes important marked based financial institutions such as security broker&dealers, finance companies, asset backed security (ABS) issuers, and commercial banks as Adrian and Shin (2009) suggest. They argue that the balance sheet fluctuations of these institutions are important determinants of real fluctuations.
Data (FRED) of St. Louis FED. I use quarterly inflation rate computed using GDP deflator to make nominal deposit rates real.
Quarterly financial data are taken from the Flow of Funds Accounts (FFA) of the Federal Reserve Board. Quarterly real data exceptHours and deposit rate data are taken from Federal Reserve Economic Data (FRED) of St. Louis FED. Hours data are taken from Current Em-ployment Statistics survey published by the Bureau of Labor Statistics. The return to capital data are taken from Gomme et al. (2011). This paper constructs an empirical measure of the return to capital for the U.S., which directly corresponds to the definition of the return to capital in this paper. The balance sheet data in the level tables of FFA are nominal and are not seasonally adjusted. All financial data are seasonally adjusted using Census X12 and are deflated using GDP deflator. I use FFA coded level tables released on March 10, 2011 when I refer to the balance sheet items of financial sector. Financial and real data sources for figures 1 and 2, and tables 1 and 2 are given below.
Liabilities are the sum of “Total liabilities” of each of the following financial institutions:
U.S. chartered commercial banks (Table L.110, Line 23), savings institutions (Table L.114, Line 23), credit unions (Table L.115, Line 16), issuers of asset-backed securities (Table L.126, Line 11), bank holding companies (Table L.112, Line 11), security brokers and dealers (Table L.129, Line 13), finance companies (Table L.127, Line 10), property-casualty insurance companies (Table L.116, Line 16), life insurance companies (Table L.117, Line 16), funding corporations (Table L.130, Line 12), and real estate investment trusts (Table L.128, Line 11).
Net Worth is the sum of “Total financial assets” minus the sum of “Total liabilities” of each of the following financial institutions: U.S. chartered commercial banks (Table L.110, Line 1 minus Line 23), savings institutions (Table L.114, Line 1 minus Line 23), credit unions (Table L.115, Line 1 minus Line 16), issuers of asset-backed securities (Table L.126, Line 1 minus Line 11), bank holding companies (Table L.112, Line 1 minus Line 11), security brokers and dealers (Table L.129, Line 1 minus Line 13), finance companies (Table L.127, Line 1 minus Line 10), property-casualty insurance companies (Table L.116, Line 1 minus Line 16), life insurance companies (Table L.117, Line 1 minus Line 16), funding corporations (Table L.130, Line 1 minus Line 12), and real estate investment trusts (Table L.128, Line 1 minus Line 11).
Leverage Ratio is is the ratio ofLiabilities toNet Worth. Finally,Credit Spread is computed as the difference between the quarterly return to capital and the quarterly deposit rate.
Consumption is the sum of “Personal consumption expenditures on nondurables” (PCND) and “Personal consumption expenditures on services”. Investment is the sum of “Personal
con-sumption expenditures on durables” (PCDG) and “Gross private domestic investment” (GPDI).
GDP is the sum of Consumption and Investment. Hours is computed as the multiplication of
“average weekly hours in private sector” with “average number of workers in private sector”.
Table 5: Business Cycle Statistics, Quarterly U.S. Data, 1952-2009
GDP C I Leverage R. Liabilities Net Worth Credit Spread
Standard deviation (%) 1.97 0.89 5.56 5.33 2.16 5.76 0.22
Quarterly autocorrelation 0.83 0.86 0.82 0.74 0.92 0.79 0.75
GDP 1 0.54 0.96 -0.08 0.57 0.28 -0.56
C – 1 0.29 0.10 0.07 -0.08 -0.05
Correlation matrix I – – 1 -0.10 0.63 0.33 -0.62
Leverage R. – – – 1 -0.03 -0.92 0.14
Liabilities – – – – 1 0.40 -0.51
Net Worth – – – – – 1 -0.32
Credit Spread – – – – – 1
aBusiness cycle statistics for GDP, consumption and investment are computed using quarterly data from FRED database.
Consumption is the sum of personal consumption expenditures on nondurables and services (PCND + PCESV). Investment is the sum of personal consumption expenditures on durable goods and gross private domestic investment (PCDG + GPDI).
GDP is the sum of consumption and investment.
bBusiness cycle statistics in the table are based on HP-filtered cyclical components over the period 1952-2009.
cThe correlation coefficients greater than 0.13 are statistically significant at 5% significance level.
Table 5 presents business cycle statistics for the aggregate leverage ratio, aggregate liabili-ties, and aggregate equity of U.S. financial sector together with those for the credit spread. The volatility of the leverage ratio is nearly 3 times larger than that of output and is roughly equal to that of investment. Table 1 shows that the financial leverage ratio is acyclical. The contem-poraneous correlation between the financial leverage ratio and output is -0.08. The volatility of aggregate equity is 3 times larger than that of output, while the volatility of aggregate debt is roughly equal to that of output.29 The contemporaneous correlation between aggregate liabili-ties and output is 0.57 while that between aggregate equity and output is 0.28, indicating that both series are procyclical.30 Moreover, the contemporaneous correlation with between credit spread and GDP is -0.56, showing that it is countercyclical.
Table 6 displays the cross-correlations of financial variables with different lags and leads of GDP. It shows that aggregate financial variables lead business cycles in the U.S. In particular, the financial leverage ratio, equity and credit spread lead output by three, two and one quarters, respectively. However, liabilities contemporaneously move with output.
The following facts emerge from the empirical analysis above: (1) Financial leverage ratio and equity are three times more volatile than output, liabilities are a little more volatile than
29Using the Flow of Funds database, Jermann and Quadrini (2009) shows that relative volatilities of non-financial sector debt and equity to nonnon-financial business sector GDP are 1.29 and 1.05, respectively.
30Jermann and Quadrini (2009) find that debt is countercyclical and equity is procyclical for non-financial firms for the same time period. In addition, using Compustat database, Covas and Den Haan (2006) shows that debt and equity issuance is procyclical for the majority of publicly listed firms.
Table 6: Cross Correlations of Financial Variables with Lags and Leads of GDP
Variable Yt−5 Yt−4 Yt−3 Yt−2 Yt−1 Yt Yt+1 Yt+2 Yt+3 Yt+4 Yt+5
Liabilities 0.01 0.13 0.27 0.41 0.52 0.57 0.57 0.50 0.39 0.26 0.12 N etW orth 0.00 0.04 0.09 0.14 0.21 0.28 0.34 0.35 0.31 0.20 0.05 LeverageR. 0.00 0.00 0.00 0.00 -0.03 -0.08 -0.14 -0.18 -0.18 -0.10 0.00 Spread 0.28 0.17 0.03 -0.15 -0.34 -0.56 -0.67 -0.60 -0.46 -0.29 -0.11
aSee the footnote (b) in Table 2 for the construction of aggregate financial variables.
bBusiness cycle statistics in the table are based on HP-filtered cyclical components over the period 1952-2009.
cThe correlation coefficients greater than 0.13 are statistically significant at 5% significance level.
output, (2) liabilities and equity are procyclical, financial leverage ratio is acyclical, and credit spread is countercyclical, and (3) Financial leverage ratio, equity and credit spread lead output by three, two and one quarters, respectively, while liabilities contemporaneously move with output.